A397005 - OEIS

%I #5 Jun 13 2026 04:39:59

%S 1,0,1,2,7,2,7,8,8,5,2,9,7,5,0,5,4,4,4,8,7,9,5,6,1,6,4,7,8,8,8,2,7,4,

%T 4,8,7,8,4,1,5,5,7,9,7,7,9,7,7,8,9,5,1,7,1,8,2,5,3,9,8,2,6,1,1,9,9,3,

%U 0,8,5,9,8,4,9,9,8,0,8,0,6,7,5,9,0,9,7,8,3,8,0,0,2,5,6,9,6,7,8,8,9,6,1,3,7

%N Decimal expansion of Sum_{k>=1} H(k)/k^7, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

%D Bruce C. Berndt, Ramanujan's Notebooks Part 1, Springer Verlag, 1985, pp. 252-253, eq. (9.5).

%D H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, p. 228, problem 14.

%H David Borwein and Jonathan M. Borwein, <a href="https://doi.org/10.1090/S0002-9939-1995-1231029-X">On an intriguing integral and some series related to zeta(4)</a>, Proc. Amer. Math. Soc., Vol. 123, No. 4 (1995), pp. 1191-1198. See p. 1196.

%H Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/477">Meditationes circa singulare serierum genus</a>, Novi Commentarii academiae scientiarum Petropolitanae, Vol. 20 (1776), pp. 140-186.

%H C. Georghiou and A. N. Philippou, <a href="https://www.fq.math.ca/Scanned/21-1/georghiou.pdf">Harmonic sums and the Zeta function</a>, Fibonacci Quarterly, Vol. 21, No. 1 (1983), pp. 29-36.

%H Ross C. McPhedran and David H. Bailey, <a href="https://arxiv.org/abs/2311.06294">New Results for Euler Sums</a>, arXiv:2311.06294 [math.NT], 2023-2025.

%H Cornel Ioan Vălean, <a href="https://doi.org/10.1007/978-3-030-02462-8">(Almost) Impossible Integrals, Sums, and Series</a>, Springer International Publishing, 2019, section 3.10, p. 87, eq. (3.45).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>, eq. (24).

%H <a href="/index/Ha#harmonic">Index entries for sequences related to harmonic numbers</a>.

%F Equals 9*zeta(8)/4 - zeta(3)*zeta(5).

%e 1.012727885297505444879561647888274487841557977977895...

%t RealDigits[9*Zeta[8]/4 - Zeta[3]*Zeta[5], 10, 120][[1]]

%o (PARI) 9*zeta(8)/4 - zeta(3)*zeta(5)

%Y Cf. A001008, A002805.

%Y Cf. A002117, A013663, A013666.

%Y Sum_{k>=1} H(k)/k^m: A076788 (m=1), A233090 (m=2), A233033 (m=3), A396737 (m=4), A397003 (m=5), A397004 (m=6), this constant (m=7), A397006 (m=8), A397007 (m=9), A397008 (m=10).

%K nonn,cons

%O 1,4

%A _Amiram Eldar_, Jun 13 2026